Network Curvature, friendship paradox and dispersion Rik Sarkar - - PowerPoint PPT Presentation

Network Curvature, friendship paradox and dispersion

Rik Sarkar

Recap: Hyperbolic distances

• Points in a disk
• Shortest paths along circular

curves bent toward the center

• Similar to internet paths being

bent toward the core

• Distances look cramped close

to the boundaries

Internet emulates hyperbolic metrics

• Shavitt, Tankel. ACM ToN 2008.

Hyperbolic model for networks

• People connect to popular “central” nodes
• Preferential attachment. Hubs. Cause small diameters.
• People connect to other “similar” nodes
• Similar in location, or interests, or communities
• Similar means small distance in some measure
• Preferential attachment does not model this well
• Cannot model the clustering properties

Popularity/similarity model

• Put all nodes on the plane at polar coord: (r, θ)
• Popularity: Distance from the center
• Like preferential attachment, earlier nodes are popular
• If a node appears at time t, its distance from center is r = ln t
• Interests/features for similarity: Represented by angle
• θ
• Two nodes a,b are similar if |θa - θb| is small.

Edge attachments

• A new node appears at

time t

• Sets r = ln t
• Sets θ = random
• It connects to the k

nearest nodes in hyperbolic distance

• Central nodes are older

and higher degree

Properties

• Creates power law

distribution

• Creates strong clustering
• Different from pref.

attachment

• More realistic in real

networks

Modeling the internet

• A suitable hyperbolic

embedding gives very good model of connection probabilities

• Similar results in other

power law networks

Actor networks

• Does not work equally

well

• Popularity vs Similarity in Growing Networks
• Papdopoulos et al. Nature 2012.

Hyperbolic geometry

• Useful in modeling metrics with exponential growth

(number of nodes within distance x)

• E.g. balanced binary tree
• Many parameters may have such properties
• Position in a hierarchy
• Topological types of paths in a domain
• Subsets of items

Few other things

Friendship paradox

• Your friends have more friends than you do!
• Are you less social than others?

Friendship paradox

• The paradox:
• If you ask everyone to report their degrees, you get the average

degree

• If you ask everyone to report the average degrees of their friends

and take the averages of all,

• you get more than the overall average degree!
• Most of us have some popular friends (hence they are popular)
• If you pick a random friend of a random person, (random edge)
• This friend is relatively likely to be popular, since popular nodes

have more edges

Friendship paradox

• Average degree of nodes:
• A node with degree d(v) contributes d(v) once
• Average degree of a friend:
• Each person picks a friend and counts degree
• A node with degree d(v) contributes d(v) times, with total

contribution d(v)2

• A few nodes with relatively high d(v) can skew the count
• https://en.wikipedia.org/wiki/Friendship_paradox
• S. L. Feld, Why your friends have more friends than you do,

American journal of sociology, 1991

Identify spouses or romantic partners

• Suppose you have the facebook graph
• Only the graph and nothing else
• Can you identify which edges correspond to

spouses or romantic partners?

Identify spouses or romantic partners

Identify spouses or romantic partners

• Tie strengths are important
• Romantic ties tend to be of high strength, more

likely to transmit information

• Do you expect romantic links to have high

embeddedness (number/fraction of common friends)?

• People have clusters of friend

circles

• Work, school, college,

hobbies

• Edges in these have high

embeddedness, even if they are not strong friends

• Spouses usually know some friends in each-others

different circles

• The edge does not have high embeddedness
• Compared to links in groups such as school/

college

• But, it has a dispersed structure:
• There are several mutual friends, but the mutual

friends are not well connected among themselves

Dispersion

• dispersion between u,v
• Notations:
• C(u,v): Common friends of u, v
• Gu : Subgraph induced by u and all neighbors of u
• duv : distance measured in Gu-{u,v}: Without using u or v

disp(u, v) = X

s,t∈C(u,v)

duv(s, t)

Dispersion

• Increases with more mutual friends
• Increases when these friends are far in the graph
• It is possible to use other distance measures
• Good results with d = 1 if no direct edge, 0 otherwise

disp(u, v) = X

s,t∈C(u,v)

duv(s, t)

Normalized dispersion

• Use norm(u,v) = disp(u,v)/embed(u,v)
• 48% accuracy
• Apply recursively, to weigh higher nodes with high dispersion
• Gives 50.5% accuracy
• 60% accuracy for married couples
• High accuracy considering hundreds of friends
• Works better than usual machine learning based on posts, visits, photos

etc features

• Best results with combination of features

• Backstrom and Kleinberg. Romantic partnerships

and dispersion of social ties, ACM CSCW 2014